Chaos in a well Effects of competing length scale

arXiv:nlin/0005035v2[nlin.CD]12Jun2001Chaosinawell:EffectsofcompetinglengthscalesR.Sankaranarayanan∗,A.Lakshminarayan†andV.B.Sheorey‡PhysicalResearchLaboratory,Navrangpura,Ahmedabad380009,India.AbstractAdiscontinuousgeneralizationofthestandardmap,whicharisesnaturallyasthedynamicsofaperiodicallykickedparticleinaonedimensionalinfi-nitesquarewellpotential,isexamined.Existenceofcompetinglengthscales,namelythewidthofthewellandthewavelengthoftheexternalfield,in-troducenoveldynamicalbehaviour.Deterministicchaosinduceddiffusionisobservedforweakfieldstrengthsasthelengthscalesdonotmatch.Thisisrelatedtoanabruptbreakdownofrotationallyinvariantcurvesandinpartic-ularKAMtori.Anapproximatestabilitytheoryisderivedwhereintheusualstandardmapisapointof“bifurcation”.PACSnumber(s):05.45Ac,45.05.+x,47.52.+jKeywords:ChaosI.INTRODUCTIONTheconstructionandstudyofareapreservingmappingshasledtoadeeperunderstand-ingofapparentlycomplexdynamics,especiallyofHamiltonianchaos.Themapsrangefromabstractmodels[1]suchasthecatmapsandthebakermaptomoregenericones.Oneofthemostwellstudiedofsuchgenericmappingsisthestandardmap[2],whichhasalsobeeninvestigatedextensivelyinitsquantumversion[3].Theclassicalmaponthecylinderdisplaysarangeofdynamicalbehaviours,andwhencompletelychaotic,diffusiverandomwalksinmomentumtakeplace.Recentexperimentsusingtrappedultra-coldsodiumatomsinpulsedlaserfieldshaveprobedthismodel,asakickedrotator,andverifiedthecentralphenomenonofquantumlocalizationofmomentumdiffusion[4].InthisLetterwestudytheclassicaldynamicsofasimplegeneralizationofthestandardmapthatnaturallyarisesfromthedynamicsofaparticletrappedinaonedimensionalin-finitelydeepwell.Thevirtueofthisgeneralizationisthatontheintroductionofacompeting∗[e-mail]†[e-mail]‡[e-mail]1lengthscalenoveldynamicalbehaviourmanifests,leadingtolargescalediffusiveprocessesevenforsmallexternalfieldstrengths.Thisplacesthestandardmapwithinafamilyofgenerallydiscontinuousareapreservingmappings;thepointswherethereiscontinuitycor-respondingtotheusualstandardmap.Westudythenoveldynamicalchangesbysimplemethodsexploitingseparationoftimescalesinnon-autonomousequations.ThefailureofthePoincar´e-Birkhofftheoremleadstotheformationofentirelystableislandsandcantoriorentirelyunstableandchaoticorbits.Itisalsotobepointedoutthatthiscanbeoneofthemethodsofcontrollingorenhancingchaos.Theclassicalmotionofaparticleinaninfinitesquarewellpotentialinthepresenceofauniformmonochromaticexternalfieldhasbeenstudiedpreviously[5].Inthiscasediffusionwasobservedoveralimitedrangeofenergiesandthereisnoissueofcompetinglengthscales.Also,noanalyticformofthestroboscopicmappingisderivable.Wehaveconsideredbelowaspacevariationoftheexternalfieldthatisalsopulsedintime,thepulsingleadingtoanalyticallyderivablemappings.Morerecentlyaspecialcaseofthissystemwasstudied[6]andquantized,where(inthechaoticregime)animportanteffectwasfound,namelydelocalizationofeigenstatesasopposedtothewellknownexponentiallylocalizedstatesofthekickedrotator.OurindependentandparallelstudiesareageneralizationandinthisLetterwepresenttheclassicalaspectsaswefeelthatthesedeserveamorecompleteunderstanding.Recentadvancesinsemiconductorphysics,makesitpossibletoconstructwellsonatomicscales.Also,developmentsininvestigatingthequantumnatureofelectronsinafinitequan-tumwellhavebeenachieved,usingtunnel-currentspectroscopy,whenthecorrespondingclassicalsystemshowschaoticbehaviour[7](seereferencestherein).Wehopethatthestudyofnoveldynamicalbehavioursinsimplemodels,asthegeneralizationofthestandardmap,willreflectonsomeaspectsofsuchexperimentallyrealizablesystems.II.KICKEDPARTICLEINAWELLWeconsideraparticleofmassm,trappedinanonedimensionalinfinitesquarewellpotentialVsq(x)ofwidth2a.ThereisanexternalfieldV(x)thatisperiodicallypulsedwithperiodT.TheHamiltonianisH=p22m+Vsq(x)+V(x)∞Xj=−∞δj−tT(1)wherex,parethepositionandmomentumoftheparticlerespectively.WeconsiderbelowV(x)=ǫcos(2πx/λ).Hereǫandλaretheamplitudeandwavelengthoftheexternalfieldrespectively.TheeffectofthepulsedfieldinHisthesameasthatofaninfinitenumberoftravellingwaveswithidenticalamplitudesandfrequencieswhicharemultiplesof2π/T(thepulsefrequency).Thestroboscopicmap,relatingthedynamicalvariablesimmediatelyaftersuccessivekicks,canbederivedinastandardmannerandresultsinthefollowingdimensionlessform:Xn+1=(−1)Mn(Xn+Pn)+(−1)Mn+1Sgn(Pn)MnPn+1=(−1)MnPn+(K/2π)sin(2πRXn+1).(2)2whereMn=hSgn(Pn)(Xn+Pn)+12i,isthenumberofbouncesoftheparticleatthewallsduringtheintervalbetweenthenthkickandthe(n+1)thkick.HereSgn(..)and[..]standforsignandintegerpartoftheargumentrespectively.ThedimensionlessquantitiesaredefinedasXn=xn2a,Pn=pnT2am,K=2ǫπ2T2amλ,R=2aλ.(3)whilethedynamicalvariablesxn,pnarethepositionandmomentumoftheparticlejustafterthenthkick.AlthoughSgn(Pn)isdiscontinuousandundefinedforPn=0,onecanseeeasilyfromthemapthateitherofthevalues(±1)canbetakenforSgn(Pn)asthisdoesnotalterthedynamics.Werefertothemapping(2)asthewellmap.NotethatKandRaretheonlytwoeffectiveparametersofthewellmap.HereKisrelatedtothestrengthoftheexternalfield;Ristheratioofthewidthofthewelltothewavele